Spherical pendulum in wind: An inexpensive laboratory-scale experiment for exploring nonlinear dynamics and estimation

This paper describes a mathematical model, estimation technique, and experiment exploring the nonlinear dynamics of a spherical pendulum subject to air drag in the presence of a steady horizontal velocity field. The pendulum’s dynamics are derived with drag as a quadratic function of the pendulum’s flow-relative velocity in both spherical and Cartesian coordinates. Numerical simulations are presented to illustrate the diverse motions that may be exhibited by the system. The model can be representative of many real-world systems that consist of an object tethered from a stationary platform in the presence of wind, or from a moving platform that induces a relative wind. The observed motion of the spherical pendulum can also be used to infer steady wind conditions. A simple experiment is described that demonstrates a pendulum-based wind sensor. A ping-pong ball is suspended on a string in the presence of an air blower and a video recording is used to measure the angular motion. The recorded measurements are then processed to infer the wind applied by the blower. Three techniques are used to compute an estimate of the wind: a static estimate based on an equilibrium assumption, a dynamic estimate that considers the instantaneous angular velocity, and an extended Kalman filtering approach. The system can serve as an instructive demonstration for students studying mechanical engineering as it provides an opportunity to discuss concepts related to damped harmonic motion, air resistance, spherical coordinates, numerical simulation, signal processing, and sensor calibration.